Vector Spaces 

INTRODUCTION 

We are familiar with two terms as the sum of vectors and multiplication of a vector by a scalar. The set of vectors forms an abelian group with respect to addition. For any three vectors u, v and w we have 

(i) Closure law : u + v is a vector 

(ii) Associative law : (u + v) + w = u + (v + w)

(iii) Existence of identity: u + 0 = 0 + u = u 

(iv) Existence of inverse : u + (-u) = (-u) + u = 0

(v) Commutative law : u+v=v+u 

- u is the additive inverse of u, is the negative of the vector u, which has the same magnitude as u but is oriented in the opposite direction. 0 is the zero vector or null vector. The multiplication of vector u by a scalar a is a vector au. Also we have 

(vi) a(bu) = (ab)u where a, b are scalars. 

(vii) a (u + v) = au + av 

(viii) (a + b)u = au +bu

We shall axiomatize the above properties by defining two operations namely internal binary operation and external binary operation. 

Let V be a non-empty set of vectors and F be a non- empty set of scalars called a field. 

The mapping o : V x V ⟹ V 

defined by u o v = u + v  ∈ V, ∀ u,v ∈ V is internal binary operation called vector addition.

The mapping o' :F x V ⟹ V defined by 

a o' u = a.u,   ∀ a ∈ F , u ∈ V is external binary operation in V over F called scalar multiplication. 

Now we shall give the abstract definition of a vector space. 

VECTOR SPACE 

Let F be a field whose elements celled scalers. Then the system (V, +, · ) consisting of 

(i) a non-empty set V of vectors,

(ii) an internal binary operation + in V called addition of vectors

(iii) an external binary operation (·) in V over F called scalar multiplication, is said to be a vector space or linear space V over F if the following axioms are satisfied : 

V₁) The system (V, +) is an abelian group i.e. (i) v₁ + v₂ ∈ V,  ∀ v₁ v₂ ∈ V

(ii) ( v₁+ v₂) + v₃ = v₁ + (v₂+ v₃)  ∀ v₁, v₂, v₃ ∈ V 

(iii) ∃ an element 0 ∈ V such that 

v+0 = 0+v= v, ∀  v ∈ V, 

The additive identity 0 is called zero vector or null vector. 

(iv) v₁ + (-v₁) = (-v₁) + v₁ = 0 

(v) v₁ + v₂ = v₂ + v₁

V₂) (i) ∀  α ∈ F, v ∈ V, α v ∈ V 

(ii) Scalar multiplication is associative i.e. 

∀  α,β ∈ F,  v ∈ V, 

α (βv) = (αβ)v 

(iii) 1. v = v ∀  v ∈ V, where 1 is the unit element of F. 

V₃) (i) Multiplication by a scalar is left distributive w.r.t. vector addition i.e. 

α(v₁+v₂) = αv₁ + αv₂, ∀ a ∈ F, v₁,v₂ ∈ V 

(ii) multiplication by a vector w.r.t. scalar addition is right distributive i.e. 

(α+β)v = αv+βv ∀ α,β ∈ F, v∈V 

The vector space V over a field F is denoted by V (F) or simply by V. 

Example 1. The set of all vectors in three dimensional space over the field R of real numbers is a vector space. 

Example 2. The field C of complex numbers is a vector space over the field R of real numbers. 

Solution. 

V₁) The system (C, +) is an abelian group. 

V₂) (i)  ∀ a ∈ R, u ∈ C, au ∈ C. 

(ii) Scalar multiplication is associative i.e. 

∀ α, β ∈ R,  u=x+iy ∈ C

α (β u) = α (β x + iβ y) = (αβ x + i αβ y) = (αβ)(x +iy) = ( αβ)u

(iii) 1u =u ∀ u∈C, where 1 is the unit element of R. 

V₃)  (i) ∀ a ∈ R,  u₁,u₂ ∈ C 

α (u₁ + u₂) = α u₁ ₊ α u₂ 

(ii) ∀ a, β ∈ R, u = C 

(α + β)u =α u +β u 

Hence C is a vector space over R.

Example 3. 

Let F be a field and H be a subfield of F. Then F forms a vector space over H w.r.t. addition in F as vector addition and the scalar product of α ∈ H and v ∈ F as the product of α and v.

Solution.

V₁) (F, +) is an abelian group. 

V₂) (i) ∀ α, β ∈ H, v ∈ F, α v ∈ F (since α ∈ F as H ⊆ F) 

(ii) since the multiplication in F is associative and so 

∀ α, β ∈ H, v ∈ F 

α ( β v) = ( α β)v 

(iii) 1 ∈ H, v ∈ F 

1.v = v 

V₃) distributive laws hold in field F. 

Hence F forms a vector space over H. 

Thus the field R of real numbers will form a vector space over the field Q of rational numbers. 

Example 4. Let F be a field. Then the set 

Fⁿ = {(a₁, a₂, ....... aₙ) : aᵢ ∈ F}

forms a vector space over F w.r.t. addition and scalar multiplication defined as follows. 

If x = (a₁, a₂, ....... aₙ), y = (b₁, b₂, ....... bₙ) ∈ Fⁿ then x + y = (a₁ + b₁, a₂ + b₂.......... aₙ + bₙ) and if a is any element of F, then 

ax = (aa₁, aa₂, .......aaₙ). 

Solution. 

V₁) The system (F , +) is an abelian group, since 

(i) ∀ x = (a₁, a₂, ....... aₙ), y = (b₁, b₂, ....... bₙ) ∈ Fⁿ

x + y = (a₁ + b₁, a₂ + b₂....... aₙ + bₙ)∈ Fⁿ

(ii) ∀ x = (a₁, a₂, ..... aₙ), y = (b₁, b₂, .... bₙ) , z= (c₁, c₂, ....cₙ) ∈ Fⁿ

(x + y) + z = (a₁ + b₁, a₂ + b₂...... aₙ + bₙ) + (c₁, c₂, ....cₙ)

= (a₁+b₁)+c₁, (a₂+b₂)+c₂, ......(aₙ+bₙ)+cₙ

=a₁+(b₁+c₁), a₂+(b₂+c₂), .......aₙ+(bₙ+cₙ)

=x+y+z

(iii) 0=(0,0,.....,0) ∈ Fⁿ is additive identity in Fⁿ, since ∀ x ∈ Fⁿ

x+0=x 

(iv) To each x = (a₁,a₂,a₃ ... aₙ) ∈ Fⁿ, ∃ 

-x = (-a₁, - a₂, .... - aₙ) ∈ Fⁿ s.t

x + (-x) = ( 0, 0, ......0)

(v) x + y= ( a₁ + b₁, a₂ + b₂, ........aₙ + bₙ)

= (b₁ + a₁, b₂ + a₂, ....bₙ +aₙ)

= y +x, ∀ x,y ∈ Fⁿ

V₂) ∀ a ∈ F, x ∈ Fⁿ

ax = (aa₁, aa₂, .......aaₙ) ∈ Fⁿ

(ii) a,b ∈ F, x ∈ Fⁿ

a(bx) = a(ba₁, ba₂, .....baₙ)

=(aba₁, aba₂, ......abaₙ)

=(ab)x

(iii) ∀ x ∈ Fⁿ

1.x = x where 1 is the unit element of F,

V₃) (i) ∀ a ∈ F, x,y ∈ Fⁿ

a(x + y)= a ( a₁ + b₁, a₂ + b₂, ........aₙ + bₙ)

= ( aa₁ + ab₁, aa₂ + ab₂, ........aaₙ + abₙ)

= ax + ay

(ii) ∀ a,b ∈ F, x ∈ Fⁿ

(a + b)x =( ( a + b)a₁, (a + b)a₂, ........(a + b)aₙ)

= ( aa₁ + ba₁, aa₂ + ba₂, ....., aaₙ + baₙ)

= ax + bx

Hence Fⁿ is a vector space over F. 

Fⁿ is called an n-dimensional vector space over the field F. 

Example 5. (i) The set Qⁿ of all ordered n tuples of rational numbers is the rational vector space over the field Q of rational numbers. 

(ii) The set Rⁿ of all ordered n tuplcs of real numbers is real vector space over the real field R. Rⁿ is also called an n-dimensional real coordinate space. 

(iii) The set Cⁿ of all ordered n tuples of complex numbers forms the complex vector space over the complex field C. 

Example 6. A vector space consisting of a single clement, namely, the zero vector 0, is called zero space or null space and is denoted by {0}.